ESE 531: Digital Signal Processing

Transcription

1 ESE 531: Digital Signal Processing Lec 10: February 14th, 2017 Practical and Non-integer Sampling, Multirate Sampling

2 Lecture Outline! Downsampling/Upsampling! Practical Interpolation! Non-integer Resampling! Multi-Rate Processing " Interchanging Operations! Polyphase Decomposition! Multi-Rate Filter Banks 2

3 Downsampling! Definition: Reducing the sampling rate by an integer number 3

4 Downsampling 4

5 Example 2π 4π 5

6 Example 2π 4π 6π 6

7 Example 7

8 Example 8

9 Upsampling! Definition: Increasing the sampling rate by an integer number x[n] = x c (nt ) x i [n] = x c (nt ') 9

10 Upsampling x i [n] 10

11 Frequency Domain Interpretation 11

12 Frequency Domain Interpretation 12

13 Example 13

14 Example 14

15 Example 15

16 Example 16

17 Example 17

18 Example 18

19 Practical Interpolation! Interpolate with simple, practical filters " Linear interpolation samples between original samples fall on a straight line connecting the samples " Convolve with triangle instead of sinc 19

20 Practical Interpolation! Interpolate with simple, practical filters " Linear interpolation samples between original samples fall on a straight line connecting the samples " Convolve with triangle instead of sinc 20

21 Frequency Domain Interpretation 21

22 Linear Interpolation -- Frequency Domain x i [n] = x e [n] h lin [n] LPF approx 22

23 Linear Interpolation -- Frequency Domain x i [n] = x e [n] h lin [n] LPF approx 23

24 Linear Interpolation -- Frequency Domain x i [n] = x e [n] h lin [n] LPF approx 24

25 Non-integer Sampling! T =TM/L " Upsample by L, then downsample by M interpolator decimator 25

26 Non-integer Sampling! T =TM/L " Upsample by L, then downsample by M interpolator decimator 26

27 Example! T =3/2T # L=2, M=3 27

28 Example! T =3/2T # L=2, M=3 28

29 Non-integer Sampling! T =TM/L " Downsample by M, then upsample by L? interpolator decimator 29

30 Multi-Rate Signal Processing! What if we want to resample by 1.01T? " Expand by L=100 " Filter π/101 ($$$$$) " Downsample by M=101! Fortunately there are ways around it! " Called multi-rate " Uses compressors, expanders and filtering 30

31 Interchanging Operations Upsampling -expanding in time -compressing in frequency Downsampling -compressing in time -expanding in frequency 31

32 Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency? 32

33 Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency? 33

34 Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency 34

35 Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency = 35

36 Interchanging Operations - Compressor Downsampling -compressing in time -expanding in frequency = 36

37 Interchanging Operations - Compressor = 37

38 Interchanging Operations - Compressor = = 38

39 Interchanging Operations - Compressor = = 39

40 Interchanging Operations - Compressor = = After compressing 40

41 Interchanging Operations - Summary Filter and expander Expander and expanded filter* Compressor and filter Expanded filter* and compressor *Expanded filter = expanded impulse response, compressed freq response 41

42 Multi-Rate Signal Processing! What if we want to resample by 1.01T? " Expand by L=100 " Filter π/101 ($$$$$) " Downsample by M=101! Fortunately there are ways around it! " Called multi-rate " Uses compressors, expanders and filtering 42

43 Polyphase Decomposition! We can decompose an impulse response (of our filter) to: 43

44 Polyphase Decomposition! We can decompose an impulse response (of our filter) to: 44

45 Polyphase Decomposition 45

46 Polyphase Decomposition 46

47 Polyphase Decomposition 47

48 Polyphase Decomposition 48

49 Polyphase Decomposition 49

50 Polyphase Implementation of Decimation! Problem: " Compute all y[n] and then throw away -- wasted computation! " For FIR length N # N mults/unit time 50

51 Polyphase Implementation of Decimation 51

52 Polyphase Implementation of Decimation 52

53 Interchanging Operations - Summary Filter and expander Expander and expanded filter Compressor and filter Expanded filter and compressor 53

54 Polyphase Implementation of Decimation 54

55 Polyphase Implementation of Decimation Each filter computation: -N/M multiplications -1/M rate per sample #N/M*(1/M) mults/unit time Total computation: -M filters #N/M mults/unit time 55

56 Multi-Rate Signal Processing! What if we want to resample by 1.01T? " Expand by L=100 " Filter π/101 ($$$$$) " Downsample by M=101! Fortunately there are ways around it! " Called multi-rate " Uses compressors, expanders and filtering 56

57 Polyphase Implementation of Decimator interpolator decimator 57

58 Polyphase Implementation of Interpolation interpolator decimator E 0 (z) E 0 (z) E 0 (z) 58

59 Multi-Rate Filter Banks! Use filter banks to operate on a signal differently in different frequency bands " To save computation, reduce the rate after filtering 59

60 Multi-Rate Filter Banks! Use filter banks to operate on a signal differently in different frequency bands " To save computation, reduce the rate after filtering! h 0 [n] is low-pass, h 1 [n] is high-pass " Often h 1 [n]=e jπn h 0 [n] $ shift freq resp by π 60

61 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass 61

62 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass 62

63 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass 63

64 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass Have to be careful with order! 64

65 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass 65

66 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass 66

67 Multi-Rate Filter Banks! h 0, h 1 are NOT ideal low/high pass 67

68 Non Ideal Filters! h 0, h 1 are NOT ideal low/high pass 68

69 Non Ideal Filters 69

70 Perfect Reconstruction non-ideal Filters 70

71 Quadrature Mirror Filters Quadrature mirror filters 71

72 Big Ideas! Downsampling/Upsampling! Practical Interpolation! Non-integer Resampling! Multi-Rate Processing " Interchanging Operations! Polyphase Decomposition! Multi-Rate Filter Banks 72

73 Admin! HW 4 due Friday " Typo in code in MATLAB problem, corrected handout " See Piazza for more information 73